# Ring ideal

In abstract algebra, an ideal is a subset I of a ring R which is closed under the ring operations in the following sense:

• for any a, b in I, we have a + b in I
• for any a in I and r in R, we have ra in I and ar in I.

If the ring is not commutative, these ideals are sometimes called two-sided to distinguish them from the left-sided (where only ra in I is required in the second condition) and the right-sided ideals. The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules of this module. In commutative rings, the three concepts coincide.

### Examples

• The even integers form an ideal in the ring Z of all integers.
• The set of all polynomials with real coefficients which are divisible by the polynomial x2 + 1 is an ideal in the ring of all polynomials.
• The set of all n-by-n matrices whose last column is zero forms a left-sided ideal in the ring of all n-by-n matrices.
• The ring C(R) of all continuous functions f : R -> R contains the ideal of all continuous functions f with f(1) = 0.
• {0} and R are ideals in every ring R.

### Factor rings and kernels

Ideals are important because they appear as the kernels of ring homomorphisms and allow to define factor rings, as will be described next.

If f : R -> S is a ring homomorphism, i.e. a function with f(a + b) = f(a) + f(b), f(ab) = f(a) f(b) for all a, b in R and f(1) = 1, then the kernel of f is defined as

ker(f) = {a in R : f(a) = 0}

The kernel is always a two-sided ideal of R.

Conversely, if we start with a two-sided ideal I of R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if b - a is in I. In case a ~ b, we say that "a and b are congruent modulo I". The equivalence class of the element a in R is given by

a + R = {a + r : r in R}

The set of all equivalence classes is denoted by R/I; it turns into a ring, the factor ring of R by I, if one defines

• (a + R) + (b + R) = (a + b) + R and
• (a + R) * (b + R) = (ab) + R.

The map p : R -> R/I defined by p(a) = a + R is a surjective ring homomorphism with kernel equal to I.

An ideal I is called proper if IR; it is called maximal if the only proper ideal it is contained in is itself. Every ideal is contained in a maximal ideal, a consequence of Zorn's lemma. If R is commutative and I is a maximal ideal, then R/I is a field. The only ideals in a field are {0} and the field itself.

### Ideal operations

The sum and the intersection of ideals is again an ideal; with these two operations, the set of all ideals of a given ring forms a lattice.

If A is any subset of the ring R, we can define the ideal generated by A to be the smallest ideal of R containing A; it is denoted by <A> and contains all finite sums of the form ∑ riaisi with ri and si in R and ai in A. The product of two ideals I and J is defined to be the ideal IJ generated by all products of the form ab with a in I and b in J. It is contained in the intersection of I and J.